Question

Which of the following statements is true for the function:

\[ f(x) = \begin{cases} x^2 + 3, & \text{if } x \neq 0, \\ 1, & \text{if } x = 0. \end{cases} \]

• A. \( f(x) \) is continuous and differentiable \( \forall x \in \mathbb{R} \)
• B. \( f(x) \) is continuous \( \forall x \in \mathbb{R} \)
• C. \( f(x) \) is continuous and differentiable \( \forall x \in \mathbb{R} - \{0\} \)
• D. \( f(x) \) is discontinuous at infinitely many points

Hint:

Check for continuity by comparing the left-hand limit, right-hand limit, and function value at the point of interest. Differentiability implies continuity.

Answer 1

The Correct Option is C

Step 1: Check continuity at \( x = 0 \). \[ \lim_{x \to 0} f(x) = \lim_{x \to 0} (x^2 + 3) = 3. \] Since \( f(0) = 1 \), and \( \lim_{x \to 0} f(x) \neq f(0) \), the function is not continuous at \( x = 0 \). 
Step 2: Check differentiability at \( x = 0 \). Since continuity is a prerequisite for differentiability, \( f(x) \) is not differentiable at \( x = 0 \). 
Step 3: Analyze for \( x \neq 0 \). For \( x \neq 0 \), \( f(x) = x^2 + 3 \), which is a polynomial function, hence it is both continuous and differentiable. Final Answer: \[ \boxed{\text{(c) } f(x) \text{ is continuous and differentiable } \forall x \in \mathbb{R} - \{0\}} \]